Wavelets in Applied and Pure Mathematics
Postgraduate Course in Applied Analysis

Vladimir V. Kisil

Web page: http://maths.leeds.ac.uk/~kisilv/courses/wavelets.html

Description: The course gives an overview of wavelets (or coherent states) construction and its realisations in applied and pure mathematics. After a short introduction to wavelets based on the representation theory of groups we will consider:

• Spaces of analytic functions with reproducing kernels: the Hardy and the Bergman spaces, etc.;

• The Fock-Segal-Bargmann space and Berezin-Toeplitz quantisation;

• Functional calculus of self-adjoint operators;

• Elements of signal processing.

The variety of applications is essentially grouped just around three groups: the Heisenberg group, SL₂(R), and ax+b group.

This course is suitable for postgraduate students in both applied and pure mathematics.

For on-line reading lecture notes are available in PDF (recommended!) or HTML formats. See Technical notes on viewing online materials. To print lecture notes use PostScript.
Contents:

1. What Are Wavelets and What Are They Good for?

2. Groups and Homogeneous Spaces.

3. Representation Theory.

4. Wavelets on Groups and Square Integrable Representations.

5. Wavelets on Homogeneous Spaces: the Segal-Bargmann Space.

6. Wavelets in Banach Spaces and Functional Calculus

Pre-requisites: Basic real and complex analysis, rudimentary algebra.

References

[1]

Syed Twareque Ali, Jean-Pierre Antoine, and Jean-Pierre Gazeau. Coherent states, wavelets and their generalizations. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 2000.

[2]

Michael E. Taylor. Noncommutative Harmonic Analysis, volume 22 of Math. Surv. and Monographs. American Mathematical Society, Providence, R.I., 1986.

• If you experienced a problem reading the course HTML pages then an easy solution could be found in Help on Browser Problems or (even better) use the PDF files (see bellow).

• PDF version is provided online and preferable in many respects. Acrobat Reader for your computer platform could be obtained free of charge.

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• These pages are created using free software from L^AT_EX  source, there are also few advises on Web publishing.

• Send comments and remarks by email kisilv@maths.leeds.ac.uk.